Questions involving differentials (again)

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What is the change of variables using differentials trick K&K are referring to here?

http://books.google.com.au/books?id=Hmqvhu7s4foC&pg=PA153&lpg=PA153&dq=kleppner+change+of+variables+differentials+intractable&source=bl&ots=Fhk8aKe7wM&sig=MJ5LPDJU98rDpsldtVPewznUIQk&hl=en#v=onepage&q=kleppner change of variables differentials intractable&f=false

(about halfway down the page)

Are there any formalities behind this?

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Also, when people derive the kinematic equations using calculus? I notice they rely on differentials

e.g.
http://physics.info/kinematics-calculus/

The first one, they had a=dv/dt then multiplied both sides by dt and integrated with respect to that variable...perhaps it's cause I'm still not all that comfortable with playing around with differentials like that yet but it doesn't seem 'proper' to do that. Are there alternate methods that DON'T involve treating differentials like that?

Another method that cancelled the differentials is shown here at the end:
I'm not sure about that either
 
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haruspex
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Oh yes, just found it. It looks like the substitution rule for integration...

For the second part (kinematic equations link), when they integrate the differential, don't integral signs already come with the differential, the variable that you're integrating with respect to?
 
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haruspex
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For the second part (kinematic equations link), when they integrate the differential, don't integral signs already come with the differential, the variable that you're integrating with respect to?
Are you referring to the dv = a.dt line? That is just saying that in a small interval of time, dt, the velocity increase, dv, will be a.dt. This is the logical first step whether you're integrating or differentiating. From there, you can either divide both sides by dt, then take the limit as dt tends to zero, to get the derivative; or perform a sum of dt's over a range, then take the limit to obtain an integral.
Does that help?
 
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^ Yes! Thanks a lot!
 

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